ThmDex – An index of mathematical definitions, results, and conjectures.

Symmetry of simple mutual information

Formulation 0

Let $X : \Omega \to \mathcal{X}$ and $Y : \Omega \to \mathcal{Y}$ each be a D5723: Simple random variable.
Let $a \in (0, \infty) \setminus \{ 1 \}$ be a D5407: Positive real number.

Then \begin{equation} I(X ; Y) = I(Y ; X) \end{equation}

Proofs

Proof 0

Let $X : \Omega \to \mathcal{X}$ and $Y : \Omega \to \mathcal{Y}$ each be a D5723: Simple random variable.
Let $a \in (0, \infty) \setminus \{ 1 \}$ be a D5407: Positive real number.

This result follows directly from the definition using result R4041: Commutativity of finite basic real summation for two sums. $\square$